// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012-2016 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
// Copyright (C) 2016 Tobias Wood <tobias@spinicist.org.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_GENERALIZEDEIGENSOLVER_H
#define EIGEN_GENERALIZEDEIGENSOLVER_H

#include "./RealQZ.h"

namespace Eigen {

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class GeneralizedEigenSolver
  *
  * \brief Computes the generalized eigenvalues and eigenvectors of a pair of general matrices
  *
  * \tparam _MatrixType the type of the matrices of which we are computing the
  * eigen-decomposition; this is expected to be an instantiation of the Matrix
  * class template. Currently, only real matrices are supported.
  *
  * The generalized eigenvalues and eigenvectors of a matrix pair \f$ A \f$ and \f$ B \f$ are scalars
  * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda Bv \f$.  If
  * \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and
  * \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V =
  * B V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
  * have \f$ A = B V D V^{-1} \f$. This is called the generalized eigen-decomposition.
  *
  * The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the
  * matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is
  * singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex \f$ \alpha \f$
  * and real \f$ \beta \f$ such that: \f$ \lambda_i = \alpha_i / \beta_i \f$. If \f$ \beta_i \f$ is (nearly) zero,
  * then one can consider the well defined left eigenvalue \f$ \mu = \beta_i / \alpha_i\f$ such that:
  * \f$ \mu_i A v_i = B v_i \f$, or even \f$ \mu_i u_i^T A  = u_i^T B \f$ where \f$ u_i \f$ is
  * called the left eigenvector.
  *
  * Call the function compute() to compute the generalized eigenvalues and eigenvectors of
  * a given matrix pair. Alternatively, you can use the
  * GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool) constructor which computes the
  * eigenvalues and eigenvectors at construction time. Once the eigenvalue and
  * eigenvectors are computed, they can be retrieved with the eigenvalues() and
  * eigenvectors() functions.
  *
  * Here is an usage example of this class:
  * Example: \include GeneralizedEigenSolver.cpp
  * Output: \verbinclude GeneralizedEigenSolver.out
  *
  * \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
  */
template <typename _MatrixType> class GeneralizedEigenSolver
{
public:
    /** \brief Synonym for the template parameter \p _MatrixType. */
    typedef _MatrixType MatrixType;

    enum
    {
        RowsAtCompileTime = MatrixType::RowsAtCompileTime,
        ColsAtCompileTime = MatrixType::ColsAtCompileTime,
        Options = MatrixType::Options,
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    /** \brief Scalar type for matrices of type #MatrixType. */
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3

    /** \brief Complex scalar type for #MatrixType. 
      *
      * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
      * \c float or \c double) and just \c Scalar if #Scalar is
      * complex.
      */
    typedef std::complex<RealScalar> ComplexScalar;

    /** \brief Type for vector of real scalar values eigenvalues as returned by betas().
      *
      * This is a column vector with entries of type #Scalar.
      * The length of the vector is the size of #MatrixType.
      */
    typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> VectorType;

    /** \brief Type for vector of complex scalar values eigenvalues as returned by alphas().
      *
      * This is a column vector with entries of type #ComplexScalar.
      * The length of the vector is the size of #MatrixType.
      */
    typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ComplexVectorType;

    /** \brief Expression type for the eigenvalues as returned by eigenvalues().
      */
    typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar, Scalar>, ComplexVectorType, VectorType> EigenvalueType;

    /** \brief Type for matrix of eigenvectors as returned by eigenvectors(). 
      *
      * This is a square matrix with entries of type #ComplexScalar. 
      * The size is the same as the size of #MatrixType.
      */
    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType;

    /** \brief Default constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via EigenSolver::compute(const MatrixType&, bool).
      *
      * \sa compute() for an example.
      */
    GeneralizedEigenSolver() : m_eivec(), m_alphas(), m_betas(), m_valuesOkay(false), m_vectorsOkay(false), m_realQZ() {}

    /** \brief Default constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa GeneralizedEigenSolver()
      */
    explicit GeneralizedEigenSolver(Index size)
        : m_eivec(size, size), m_alphas(size), m_betas(size), m_valuesOkay(false), m_vectorsOkay(false), m_realQZ(size), m_tmp(size)
    {
    }

    /** \brief Constructor; computes the generalized eigendecomposition of given matrix pair.
      * 
      * \param[in]  A  Square matrix whose eigendecomposition is to be computed.
      * \param[in]  B  Square matrix whose eigendecomposition is to be computed.
      * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
      *    eigenvalues are computed; if false, only the eigenvalues are computed.
      *
      * This constructor calls compute() to compute the generalized eigenvalues
      * and eigenvectors.
      *
      * \sa compute()
      */
    GeneralizedEigenSolver(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true)
        : m_eivec(A.rows(), A.cols()), m_alphas(A.cols()), m_betas(A.cols()), m_valuesOkay(false), m_vectorsOkay(false), m_realQZ(A.cols()), m_tmp(A.cols())
    {
        compute(A, B, computeEigenvectors);
    }

    /* \brief Returns the computed generalized eigenvectors.
      *
      * \returns  %Matrix whose columns are the (possibly complex) right eigenvectors.
      * i.e. the eigenvectors that solve (A - l*B)x = 0. The ordering matches the eigenvalues.
      *
      * \pre Either the constructor 
      * GeneralizedEigenSolver(const MatrixType&,const MatrixType&, bool) or the member function
      * compute(const MatrixType&, const MatrixType& bool) has been called before, and
      * \p computeEigenvectors was set to true (the default).
      *
      * \sa eigenvalues()
      */
    EigenvectorsType eigenvectors() const
    {
        eigen_assert(m_vectorsOkay && "Eigenvectors for GeneralizedEigenSolver were not calculated.");
        return m_eivec;
    }

    /** \brief Returns an expression of the computed generalized eigenvalues.
      *
      * \returns An expression of the column vector containing the eigenvalues.
      *
      * It is a shortcut for \code this->alphas().cwiseQuotient(this->betas()); \endcode
      * Not that betas might contain zeros. It is therefore not recommended to use this function,
      * but rather directly deal with the alphas and betas vectors.
      *
      * \pre Either the constructor 
      * GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function
      * compute(const MatrixType&,const MatrixType&,bool) has been called before.
      *
      * The eigenvalues are repeated according to their algebraic multiplicity,
      * so there are as many eigenvalues as rows in the matrix. The eigenvalues 
      * are not sorted in any particular order.
      *
      * \sa alphas(), betas(), eigenvectors()
      */
    EigenvalueType eigenvalues() const
    {
        eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
        return EigenvalueType(m_alphas, m_betas);
    }

    /** \returns A const reference to the vectors containing the alpha values
      *
      * This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
      *
      * \sa betas(), eigenvalues() */
    ComplexVectorType alphas() const
    {
        eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
        return m_alphas;
    }

    /** \returns A const reference to the vectors containing the beta values
      *
      * This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
      *
      * \sa alphas(), eigenvalues() */
    VectorType betas() const
    {
        eigen_assert(m_valuesOkay && "GeneralizedEigenSolver is not initialized.");
        return m_betas;
    }

    /** \brief Computes generalized eigendecomposition of given matrix.
      * 
      * \param[in]  A  Square matrix whose eigendecomposition is to be computed.
      * \param[in]  B  Square matrix whose eigendecomposition is to be computed.
      * \param[in]  computeEigenvectors  If true, both the eigenvectors and the
      *    eigenvalues are computed; if false, only the eigenvalues are
      *    computed. 
      * \returns    Reference to \c *this
      *
      * This function computes the eigenvalues of the real matrix \p matrix.
      * The eigenvalues() function can be used to retrieve them.  If 
      * \p computeEigenvectors is true, then the eigenvectors are also computed
      * and can be retrieved by calling eigenvectors().
      *
      * The matrix is first reduced to real generalized Schur form using the RealQZ
      * class. The generalized Schur decomposition is then used to compute the eigenvalues
      * and eigenvectors.
      *
      * The cost of the computation is dominated by the cost of the
      * generalized Schur decomposition.
      *
      * This method reuses of the allocated data in the GeneralizedEigenSolver object.
      */
    GeneralizedEigenSolver& compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true);

    ComputationInfo info() const
    {
        eigen_assert(m_valuesOkay && "EigenSolver is not initialized.");
        return m_realQZ.info();
    }

    /** Sets the maximal number of iterations allowed.
    */
    GeneralizedEigenSolver& setMaxIterations(Index maxIters)
    {
        m_realQZ.setMaxIterations(maxIters);
        return *this;
    }

protected:
    static void check_template_parameters()
    {
        EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
        EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
    }

    EigenvectorsType m_eivec;
    ComplexVectorType m_alphas;
    VectorType m_betas;
    bool m_valuesOkay, m_vectorsOkay;
    RealQZ<MatrixType> m_realQZ;
    ComplexVectorType m_tmp;
};

template <typename MatrixType>
GeneralizedEigenSolver<MatrixType>& GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors)
{
    check_template_parameters();

    using std::abs;
    using std::sqrt;
    eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows());
    Index size = A.cols();
    m_valuesOkay = false;
    m_vectorsOkay = false;
    // Reduce to generalized real Schur form:
    // A = Q S Z and B = Q T Z
    m_realQZ.compute(A, B, computeEigenvectors);
    if (m_realQZ.info() == Success)
    {
        // Resize storage
        m_alphas.resize(size);
        m_betas.resize(size);
        if (computeEigenvectors)
        {
            m_eivec.resize(size, size);
            m_tmp.resize(size);
        }

        // Aliases:
        Map<VectorType> v(reinterpret_cast<Scalar*>(m_tmp.data()), size);
        ComplexVectorType& cv = m_tmp;
        const MatrixType& mS = m_realQZ.matrixS();
        const MatrixType& mT = m_realQZ.matrixT();

        Index i = 0;
        while (i < size)
        {
            if (i == size - 1 || mS.coeff(i + 1, i) == Scalar(0))
            {
                // Real eigenvalue
                m_alphas.coeffRef(i) = mS.diagonal().coeff(i);
                m_betas.coeffRef(i) = mT.diagonal().coeff(i);
                if (computeEigenvectors)
                {
                    v.setConstant(Scalar(0.0));
                    v.coeffRef(i) = Scalar(1.0);
                    // For singular eigenvalues do nothing more
                    if (abs(m_betas.coeffRef(i)) >= (std::numeric_limits<RealScalar>::min)())
                    {
                        // Non-singular eigenvalue
                        const Scalar alpha = real(m_alphas.coeffRef(i));
                        const Scalar beta = m_betas.coeffRef(i);
                        for (Index j = i - 1; j >= 0; j--)
                        {
                            const Index st = j + 1;
                            const Index sz = i - j;
                            if (j > 0 && mS.coeff(j, j - 1) != Scalar(0))
                            {
                                // 2x2 block
                                Matrix<Scalar, 2, 1> rhs =
                                    (alpha * mT.template block<2, Dynamic>(j - 1, st, 2, sz) - beta * mS.template block<2, Dynamic>(j - 1, st, 2, sz))
                                        .lazyProduct(v.segment(st, sz));
                                Matrix<Scalar, 2, 2> lhs = beta * mS.template block<2, 2>(j - 1, j - 1) - alpha * mT.template block<2, 2>(j - 1, j - 1);
                                v.template segment<2>(j - 1) = lhs.partialPivLu().solve(rhs);
                                j--;
                            }
                            else
                            {
                                v.coeffRef(j) =
                                    -v.segment(st, sz).transpose().cwiseProduct(beta * mS.block(j, st, 1, sz) - alpha * mT.block(j, st, 1, sz)).sum() /
                                    (beta * mS.coeffRef(j, j) - alpha * mT.coeffRef(j, j));
                            }
                        }
                    }
                    m_eivec.col(i).real().noalias() = m_realQZ.matrixZ().transpose() * v;
                    m_eivec.col(i).real().normalize();
                    m_eivec.col(i).imag().setConstant(0);
                }
                ++i;
            }
            else
            {
                // We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a positive diagonal 2x2 block T
                // Then taking beta=T_00*T_11, we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00):

                // T =  [a 0]
                //      [0 b]
                RealScalar a = mT.diagonal().coeff(i), b = mT.diagonal().coeff(i + 1);
                const RealScalar beta = m_betas.coeffRef(i) = m_betas.coeffRef(i + 1) = a * b;

                // ^^ NOTE: using diagonal()(i) instead of coeff(i,i) workarounds a MSVC bug.
                Matrix<RealScalar, 2, 2> S2 = mS.template block<2, 2>(i, i) * Matrix<Scalar, 2, 1>(b, a).asDiagonal();

                Scalar p = Scalar(0.5) * (S2.coeff(0, 0) - S2.coeff(1, 1));
                Scalar z = sqrt(abs(p * p + S2.coeff(1, 0) * S2.coeff(0, 1)));
                const ComplexScalar alpha = ComplexScalar(S2.coeff(1, 1) + p, (beta > 0) ? z : -z);
                m_alphas.coeffRef(i) = conj(alpha);
                m_alphas.coeffRef(i + 1) = alpha;

                if (computeEigenvectors)
                {
                    // Compute eigenvector in position (i+1) and then position (i) is just the conjugate
                    cv.setZero();
                    cv.coeffRef(i + 1) = Scalar(1.0);
                    // here, the "static_cast" workaound expression template issues.
                    cv.coeffRef(i) = -(static_cast<Scalar>(beta * mS.coeffRef(i, i + 1)) - alpha * mT.coeffRef(i, i + 1)) /
                                     (static_cast<Scalar>(beta * mS.coeffRef(i, i)) - alpha * mT.coeffRef(i, i));
                    for (Index j = i - 1; j >= 0; j--)
                    {
                        const Index st = j + 1;
                        const Index sz = i + 1 - j;
                        if (j > 0 && mS.coeff(j, j - 1) != Scalar(0))
                        {
                            // 2x2 block
                            Matrix<ComplexScalar, 2, 1> rhs =
                                (alpha * mT.template block<2, Dynamic>(j - 1, st, 2, sz) - beta * mS.template block<2, Dynamic>(j - 1, st, 2, sz))
                                    .lazyProduct(cv.segment(st, sz));
                            Matrix<ComplexScalar, 2, 2> lhs = beta * mS.template block<2, 2>(j - 1, j - 1) - alpha * mT.template block<2, 2>(j - 1, j - 1);
                            cv.template segment<2>(j - 1) = lhs.partialPivLu().solve(rhs);
                            j--;
                        }
                        else
                        {
                            cv.coeffRef(j) = cv.segment(st, sz).transpose().cwiseProduct(beta * mS.block(j, st, 1, sz) - alpha * mT.block(j, st, 1, sz)).sum() /
                                             (alpha * mT.coeffRef(j, j) - static_cast<Scalar>(beta * mS.coeffRef(j, j)));
                        }
                    }
                    m_eivec.col(i + 1).noalias() = (m_realQZ.matrixZ().transpose() * cv);
                    m_eivec.col(i + 1).normalize();
                    m_eivec.col(i) = m_eivec.col(i + 1).conjugate();
                }
                i += 2;
            }
        }

        m_valuesOkay = true;
        m_vectorsOkay = computeEigenvectors;
    }
    return *this;
}

}  // end namespace Eigen

#endif  // EIGEN_GENERALIZEDEIGENSOLVER_H
